In this work, we will answer these questions and shed more light on the response theory for non-Hermitian two-band systems by the adiabatic perturbation theory [1]}, [2]}, [3]}. The reminder of this paper is organized as follows. In Sec. , we introduce the system-environment couplings and derive a master equation by which we obtain a non-Hermitian Hamiltonian to describe the two-band system subjected to environments. The dependence of the decay rates on the Bloch vectors is also derived and discussed. In Sec. , we work out the response of non-Hermitian two-band systems to a constant electric field using the adiabatic perturbation theory. The results show that the response of the system can be divided into two terms. The first term is proportional the non-Hermitian Chern number for two-band systems, which is reminiscent of the relation between the Chern number and the Hall conductance of closed systems. While the second term can be treated as a correction that suggests the relationship between the Chern number and the conductance might fail for open systems. In Sec. , taking a tight-binding electron in a two-dimensional lattice as an example, we calculate and plot the complex-band structures as a function of the momentum of the electron, \(k_x\)  and \(k_y\) . The Hall conductance of the system is also shown and analyzed. We find that Hall conductance depends on the strength of the decay rate and the electron occupation on the Bloch band, which no longer leads to a quantized Hall conductance in general and a delay in the response of the system to the constant electric field appears. We also compare the Hall conductance of the open systems by non-Hermitian Hamiltonian and by master equation in this section. In Sec. , we conclude the paper with discussions.
